Analytical Approximations to Approximations in the Chebyshev Sense

This paper concerns approximation in the Chebyshev sense, over a prescribed interval xa ^ x ^ xb of a continuous real variable x. As 1 2 T H E BELL SYSTEM TECHNICAL J O U R N A L , JANUARY 1970 defined, ail approximation in the Chebyshev sense is a minimax approximation--one in which the maximum error is as small as is possible within given constraints on the approximating function. Minimax approximations in which errors are weighted by a prescribed function of the independent variable can also be treated as Chebyshev approximations, by multiplying the approximated and approximating functions by the weight function. Frequently, but not always, approximation in the Chebyshev sense implies an error of the "equal ripple" sort illustrated in Fig. 1--that is, a sequence of equal positive and negative extrema with monotonic variations in between. General necessary and sufficient conditions for this are not known. However, the following conditions are sufficient: the p disposable parameters of the approximating function are to be such t h a t the approximation error can be made zero at p arbitrary points within the approximating interval. Referring to Fig. 1, the arbitrary error points divide the approximation interval into p + 1 segments. There is to be a particular division such that the error function achieves its maximum magnitude p + 1 times--at the two edges of the approximation interval and once within each of the p -- 1 interior segments. There are to be no other local extrema within the approximation interval.